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The paper describes two versions of an inquiry-based activity in geometry, designed as a game between two players. The game is inspired by Hintikka's semantic game, which is a familiar tool in the field of logic to define truth. The activity is designed in a dynamic geometry environment (DGE). The inquiry is initially guided by the game itself and later by a questionnaire that helps students discover the geometry theorem behind the game. The activity is emblematic of describing a geometry-based inquiry that can be implemented with various Euclidean geometry theorems. The analysis of the first "student vs. student" version associates the example space produced by the students with their dialogue, to identify the different functions of variation. Based on the results of this version, we designed a "student vs. computer" version, and created filters for the automatic analysis of the players' moves. Our findings show that students who participated in the activity developed forms of strategic reasoning that helped them discover the winning configuration, formulate if-then statements, and validate or refute conjectures. Automation of the analysis creates new research opportunities for analyzing and assessing students' inquiry processes, and makes possible extensive experimentation on inquiry-based knowledge acquisition.

The paper analyses a basic gap, highlighted by most of the literature concerning the teaching of proofs, namely, the distance between students' argumentative and proving processes. The analysis is developed from both epistemological and cognitive standpoints: it critiques the Toulmin model of reasoning and introduces a new model, the Logic of Inquiry of Hintikka, more suitable for bridging this gap. An example of didactical activity within Dynamic Geometry Environments is sketched in order to present a concrete illustration of this approach and to show the pedagogical effectiveness of the model. Keywords Proof Á Logic of inquiry Á Argumentation Á Dynamic geometry environments 10.1 Introduction In their wonderful book Anschauliche Geometrie, Hilbert and Cohn-Vossen (1932) wrote 1 : In mathematics, as in all scientific research, we find two tendencies: the tendency to abstraction-it seeks to work out (herauszuarbeiten) the logical points of view from the manifold material and bring this into systematic connection-and the other tendency, that

Come viene messo in luce dal MIUR nel "Regolamento recante norme in materia di adempimento dell'obbligo di istruzione", tra le finalità dell'asse culturale matematico c'è anche l'acquisizione delle abilità necessarie «per seguire e vagliare la coerenza logica delle argomentazioni proprie e altrui in molteplici contesti di indagine conoscitiva e di decisione» (MIUR, 2007). In particolare, le competenze argomentative costituiscono la base per preparare gli studenti alla dimostrazione, attività che contraddistingue il pensiero matematico maturo. Garuti, Boero, Lemut e Mariotti (1996) sostengono che in alcuni casi possa esistere una "continuità cognitiva" tra i processi di produzione di congetture e la costruzione di dimostrazioni. Essa si verifica se gli studenti, nella fase di produzione della congettura, innescano un'intensa attività

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